An excerpt from Chapter 7 of my book Music: Notation and Practice in Past and Present, now in press with National Social Science Press.
The issue of how to accurately tune a musical instrument, especially one which depends on a number of independent strings, has long been a source of concern for musicians and mathematicians. One of the earliest documents of Western music is a stone tablet written in Babylonian cuneiform that has been interpreted as a set of directions for the proper tuning of a seven-string harp or lyre.
The question, then, of what tuning is and what it means to be “in tune” is central to the understanding of music and to the training of musicians. Simply put, two pitches are considered to be in tune if they have the same fundamental frequency, or if their fundamental frequencies are related to each other by a low whole-number ratio. Because human beings are by nature somewhat imprecise, both in perception and in manipulation of pitch, the practical meaning of being “in tune” is somewhat more complex. Because various cultures have accepted differing musical intervals as sounding “correct,” intonation for human ears is highly dependent on training and cultural norms, at least beyond the tuning of perfect unisons or octaves. The practice of accurately producing a desired pitch is known as intonation, and the ability to play with good intonation is a skill that takes many musicians years to master.
A useful approach may be to examine the experience of various musicians with regard to tuning in order to understand what role intonation and tuning play in the practice of music. To begin, consider a vocalist who may perform with or without accompaniment. An accomplished singer will have practiced many years to develop his “ear,” that is, his inner sense of intonation, which is usually based on an understanding of scales, keys and the relationships between notes in those keys. Most musicians do not possess the set of abilities collectively referred to as absolute pitch, also known as perfect pitch. Absolute pitch is the ability to identify a pitch by note name when it is heard or to accurately produce a requested pitch without reference to a standard. A great deal of research has been done in this area to determine just how absolute pitch is developed and expressed, and what its limitations are. It is somewhat ironic that many—if not most—musicians are unable to identify the basic materials of their art without reference to some known standard, as if a painter were able to relate colors to each other within the context of a painting, but not be able to certainly identify red as red without reference to a color wheel. A singer’s tuning and intonation, then, rely on constant comparison of notes to each other and to outside reference pitches, such as might be found in an instrumental accompaniment. It is quite possible for a person to possess an exquisite vocal instrument but to utterly lack the ability to control intonation, whether through a lack of training or a simple lack of musical aptitude. Conversely, a person may possess an inner ear, or ability to audiate musical sound, that is without peer, but be unable to control the vocal apparatus sufficiently to communicate music to others, whether through physiological structure, lack of kinesthetic ability or auditory dysfunction.
A performer on a brass instrument, then, faces a similar set of dilemmas to a vocalist, but with the additional challenge of making music using a device outside the body. The brass player relies on her training and internal ear to audiate pitches prior to playing them, but also on her knowledge of the tendencies of her instrument. Professional brass players must spend many years becoming comfortable with their personal instrument, as every trumpet, for example, is subtly different from every other trumpet, even those by the same manufacturer. The intonation tendencies of various pitches and groups of pitches are basic knowledge for a brass player. When playing in an ensemble setting, in a brass section of an orchestra, perhaps, the brass player will also be aware of the pitches played by other players and adjust her own pitch accordingly. To play a pitch in tune requires first that a player anticipate the fine adjustments that will be required, but also the ability to detect several acoustical phenomena.
For unison and octave pitches, most musicians, including brass players, utilize a technique known as beatless tuning. When two pitches are very close to one another (or close to being a perfect octave apart), the crests and troughs of their waveforms become very closely aligned, and reinforce or cancel each other in predictable patterns, resulting in beats. The closer to each other two pitches are, the slower these beats sound, and when a player detects beats, she knows to make adjustments in order to eliminate them.
Other intervals require a different approach to intonation. Beats are one aspect of an acoustical phenomenon known as difference tones. The frequency in Herz of the beats between two pitches is the arithmetic difference between the frequencies of the two pitches. Thus, when a tone is played at 440Hz simultaneously with a tone at 444Hz, the beats, or difference tones, will sound at a rate of 4Hz. When pitches are in small whole-number ratios to each other, difference tones can become quite prominent, especially to listeners in close proximity to the sound sources. For example, when two brass players play pitches a major third apart, say, A4 and C#4, when the pitches are in tune with each other, they will produce a tone reflecting the difference between their frequencies, 440 Hz and 550 Hz, respectively, for a low additional pitch of 110 Hz, or A2. Most musicians perceive this as a “buzz” that indicates that the chords have been “locked-in.” A similar phenomenon that is exploited by musicians is the summation tone, which is heard as a tone with a frequency that is the sum of the two component frequencies. In the example given above, the summation tone for 440 Hz and 550 Hz would be 990Hz, or B5. As intervals are added to each other to become chords, the achievement of summation and difference tones becomes crucial to good intonation.
At times, passagework on a brass instrument becomes too rapid for the tuning of each individual chord or interval, and a player must fall back on her experience and technique on the instrument to ensure good intonation. This comes about through years of practice and preparation, especially work on scales, arpeggios, or chords played one note at a time, and etudes, or music that incorporates common patterns or technical difficulties and is intended mainly for the practice room. Effective practice for many musicians does not mean solely the preparation of music that is to later be played in public, but rather the development of a daily routine that emphasizes the technical fundamentals for the performer’s instrument in order to be prepared for whatever music may appear in the future.
In contrast to the emphasis that vocalists or brass players must place on intonation, some instruments offer the player very little control over intonation or even none at all. The piano, for example, while certainly tunable by any musician with the proper equipment, is generally tuned prior to performance or more likely only periodically, leaving the pianist with a fixed set of pitches at his disposal. The tuning of pianos, then, is frequently left to specialized piano tuners, who may or may not be accomplished pianists, while pianists of any skill level are fairly unlikely to be skilled piano tuners.
In the era before the commercial availability of first analog and then digital tuners, piano tuning was a skill acquired only through great effort and refined through extensive experience. At one time, visually-impaired persons were frequently trained as piano tuners because it was assumed, often correctly, that the other sensory organs acquired greater acuity in those lacking the visual sense. With perfect vision not required, and the piano at one time enjoying the place in middle class homes now given over to the television, piano tuning was a logical career for persons in this situation. The difficulty in tuning a piano lies in that instrument’s use of equal temperament, in which only the octaves are tuned to the precise frequencies generated by the harmonic series. In addition, the lower strings of a piano are often tuned deliberately low, while the higher strings are tuned deliberately high, meaning that the instrument as a whole has a certain inharmonicity that has become the preferred norm in many parts of society. A final challenge is that most of the pitches on the modern piano involve not just one string but two or three, meaning that every note must be in tune with itself before it can be in tune with the rest of the instrument. Thus, while the performer at a piano may think little of intonation—except perhaps to notice that the instrument is in need of a tuning, or to notice intonation deficiencies in the performers with whom he collaborates—the process of tuning a piano is very complex and can be very time-consuming.
Other musicians are responsible for calibrating their own instruments. A common example in this regard is the guitar, either in acoustic or electric forms. Before a guitarist can play, she must tune the strings of the instrument, which on six-string models are tuned to six different pitches at varying intervals from each other. In addition, basic guitar technique often involves various alternate tunings to allow the guitarist to play certain music more easily. A guitarist will begin by tuning one string to a reference pitch, and then tune other strings, one at a time, to the first string. The frets on the neck of the guitar guarantee that one string can be tuned to another fairly easily, and many guitarists also make use of harmonics to check strings even more closely. Once the guitar is in tune, the guitarist can begin to play with relatively little attention to intonation, but she must continually listen to her instrument to insure that it hasn’t gone out of tune, and make adjustments as needed.
Both the piano and the guitar have had turns as mass-produced, popular instruments, with the piano’s heyday occurring in the 19th century, and the guitar taking the piano’s place in the late-20th century. These instruments have the advantage of being instruments that can supply complete musical textures, but can also be played while singing. A very basic technique on either instrument can supply needed accompaniment to vocal music, and both were brought into the economic reach of the European and American middle class by mass production and the rise in living standards that followed the Industrial Revolution. The piano and guitar, however, were not the only instruments to benefit from the advantages listed above. What they seem to have in common that other instruments lack is a relatively simple approach to intonation. A piano that is once tuned by a specialist and then kept within reasonable limits of temperature and humidity will remain relatively in tune for quite some time, perhaps six months or more. The guitar is easily tuned at the beginning of a session, and with high-quality strings and only minor adjustments, will remain in tune for a performance without the guitarist having to constantly worry about intonation in the manner of a brass player or violinist. This approach to intonation has certainly contributed to the intense popularity of both of these instruments.
A very few instruments are impossible to tune once manufactured and have a fixed pitch. An example of this would be so-called keyboard percussion instruments such as the xylophone, which consists of a set of hardwood or synthetic bars in graduated sizes which are struck with mallets. While it would conceivably be possible to shorten these bars, and thus raise their pitch, this is rarely done, and it would be patently impossible to lengthen the bars to lower their pitch. Thus, the percussionist can exercise no control whatsoever over intonation and is completely dependent on the skill of the manufacturer and the quality of the materials and design, which hopefully will be only minimally susceptible to changes in temperature and humidity. It is likely that other musicians will find it necessary to adjust their intonation to these instruments.
All serious musicians, and most amateurs, are able to calibrate their own instruments, with the exception of those who play piano, organ or other instruments that rely on specialists for tuning or which cannot be adjusted. When a musician performs as a soloist, with no accompaniment, all tuning and intonation can be done internally, with only the musician’s internal ear to guide the performance. A more likely situation, however, is that of the ensemble performance, in which instruments must not only be played with good intonation, but must be calibrated to each other. Since the 19th century, the international standard for ensemble performance in Western music has been to tune A4 to 440Hz, sometimes referred to as A440. Tuning forks, pitch pipes and analog and digital tuners have all come to be manufactured to this standard, as have fixed-pitch instruments, such as the keyboard percussion mentioned above. A musical ensemble, then, must develop a procedure—a ritual, even—for this calibration. The ritual calibration for an American orchestra is almost a cliché—the principal oboist, usually with the aid of a digital tuner, sounds A4, providing a standard to which the other musicians of the orchestra will then adjust their own instruments. By contrast, in Europe, orchestras do not tune onstage, with the result that if the temperature onstage is different than that offstage, the first part of the first piece to be performed may be grossly out of tune until the musicians can make the necessary tuning adjustments.
 Psychologist Carl Seashore developed a very common and very simple test for musical aptitude that measures aptitude by the ability to correctly determine which of two played pitches is higher or lower than the other.
 This may be an advantage in some senses; while the instrumentalist can inspect her instrument in great detail, a vocalist’s instrument is largely hidden from view. While the vocal folds are easily viewed using a laryngoscope, the specific skeletal, muscular and tissue structure that contributes to a fine vocal instrument are often a complete mystery, even to expert performers.
 Some musicians will also use the technique of beatless tuning for perfect fifths.
 Traditionally, piano tuning was accomplished using the phenomenon of beats—a correct equal-tempered tuning will have a certain number of beats when notes, usually “perfect” fifths, are played together. Since the development of cheap and accurate digital tuners, piano tuning in the United States has shifted from an occupation for the visually-impaired to a side-job or post-retirement business, although skilled piano technicians, who can not only tune but repair and maintain pianos, are often employed full-time by college and university music departments and urban performing arts centers.
 The modern piano is a very complex and very precise piece of technology. As a piece of engineering, it stands beside that other great achievement of the early-19th-century, the steam locomotive, in its ingenuity, and greatly surpasses the locomotive in terms of reliability, safety and longevity, as many pianos from the 1880s are still in use.
 The standard tuning of earlier times was frequently somewhat lower, and this is reflected today by the tendency of ensembles specializing in authentic 18th-century practice to tune A4 to 415Hz, or very nearly A$4 in modern calibrations.
 This sort of standardization was made possible by first the knowledge provided by the Enlightenment and the Scientific Revolution and then the mass-production of musical instruments and equipment to high tolerances and standards made possible by the Industrial Revolution. This sort of standardization of pitch would not have been practical prior to the 19th century, when local standards of pitch prevailed.
 In truth, for professional orchestras, this ritual is just that—a ritual performed for the benefit of the audience. Most musicians will have already calibrated their instruments backstage.